A3. Transition Systems and Propositional Logic

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Planning and Optimization

A3. Transition Systems and Propositional Logic

Gabriele R¨ oger and Thomas Keller

Universit¨ at Basel

October 1, 2018

G. R¨oger, T. Keller (Universit¨at Basel) Planning and Optimization October 1, 2018 1 / 27

Planning and Optimization

October 1, 2018 — A3. Transition Systems and Propositional Logic

A3.1 Transition Systems A3.2 Example: Blocks World

A3.3 Reminder: Propositional Logic A3.4 Summary

Content of this Course

Planning

Classical

Tasks Progression/

Regression Complexity Heuristics

Probabilistic

MDPs Uninformed Search

Heuristic Search Monte-Carlo

Methods

Goals for Today

Today:

I introduce a mathematical model for planning tasks:

transition systems Chapter A3

I introduce compact representations for planning tasks suitable as input for planning algorithms

Chapter A4

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A3. Transition Systems and Propositional Logic Transition Systems

A3.1 Transition Systems

Transition System Example

Transition systems are often depicted as directed arc-labeled graphs with decorations to indicate the initial state and goal states.

`

1

`

1

`

1

`

1

`

³

`

₃

`

₂

`

⁴

`

³

`

₄

`

₄

`

4

`

2

`

2

c(`

1

) = 1, c(`

2

) = 1, c (`

3

) = 5, c(`

4

) = 0

Transition Systems

Definition (Transition System)

A transition system is a 6-tuple T = hS , L, c, T , s ₀ , S _? i where

I S is a finite set of states,

I L is a finite set of (transition) labels,

I c : L → R ⁺ ₀ is a label cost function,

I T ⊆ S × L × S is the transition relation,

I s ₀ ∈ S is the initial state, and

I S _? ⊆ S is the set of goal states.

We say that T has the transition hs, `, s ⁰ i if hs , `, s ⁰ i ∈ T .

Deterministic Transition Systems

Definition (Deterministic Transition System)

A transition system is called deterministic if for all states s

and all labels `, there is at most one state s ⁰ with s − → ^` s ⁰ .

Example: previously shown transition system

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Transition System Terminology (1)

We use common terminology from graph theory:

I s ⁰ successor of s if s → s ⁰

I s predecessor of s ⁰ if s → s ⁰

Transition System Terminology (2)

We use common terminology from graph theory:

I s ⁰ reachable from s if there exists a sequence of transitions s ⁰ − ^` →

¹

s ¹ , . . . , s ⁿ⁻¹ − ^` →

ⁿ

s ⁿ s.t. s ⁰ = s and s ⁿ = s ⁰

I

Note: n = 0 possible; then s = s

⁰

I

s

⁰

, . . . , s

ⁿ

is called (state) path from s to s

⁰

I

`

₁

, . . . , `

_n

is called (label) path from s to s

⁰

I

s

⁰

−

^`

→

¹

s

¹

, . . . , s

ⁿ⁻¹

−

^`

→

ⁿ

s

ⁿ

is called trace from s to s

⁰

I

length of path/trace is n

I

cost of label path/trace is P

n i=1

c(`

_i

)

Transition System Terminology (3)

We use common terminology from graph theory:

I s ⁰ reachable (without reference state) means reachable from initial state s ₀

I solution or goal path from s : path from s to some s ⁰ ∈ S _?

I

if s is omitted, s = s

₀

is implied

I transition system solvable if a goal path from s ₀ exists

A3. Transition Systems and Propositional Logic Example: Blocks World

A3.2 Example: Blocks World

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Running Example: Blocks World

I Throughout the course, we occasionally use the blocks world domain as an example.

I In the blocks world, a number of differently blocks are arranged on a table.

I Our job is to rearrange them according to a given goal.

Blocks World Rules (1)

Location on the table does not matter.

≡

Location on a block does not matter.

≡

Blocks World Rules (2)

At most one block may be below a block.

At most one block may be on top of a block.

Blocks World Transition System for Three Blocks

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Blocks World Computational Properties

blocks states

1 1

2 3

3 13

4 73

5 501

6 4051

7 37633

8 394353 9 4596553

blocks states

10 58941091

11 824073141

12 12470162233

13 202976401213

14 3535017524403

15 65573803186921 16 1290434218669921 17 26846616451246353 18 588633468315403843

I Finding solutions is possible in linear time

in the number of blocks: move everything onto the table, then construct the goal configuration.

I Finding a shortest solution is NP-complete given a compact description of the problem.

The Need for Compact Descriptions

I We see from the blocks world example that transition systems are often far too large to be directly used as inputs

to planning algorithms.

I We therefore need compact descriptions of transition systems.

I For this purpose, we will use propositional logic, which allows expressing information about 2 ⁿ states as logical formulas over n state variables.

A3. Transition Systems and Propositional Logic Reminder: Propositional Logic

A3.3 Reminder: Propositional Logic

More on Propositional Logic

Need to Catch Up?

I This section is a reminder. We assume you are already well familiar with propositional logic.

I If this is not the case, we recommend Chapters B1 and B2 of the Theory of Computer Science course at

https://dmi.unibas.ch/de/studium/

computer-science-informatik/fs18/

main-lecture-theory-of-computer-science/

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Syntax of Propositional Logic

Definition (Logical Formula)

Let A be a set of atomic propositions.

The logical formulas over A are constructed by finite application of the following rules:

I > and ⊥ are logical formulas (truth and falsity).

I For all a ∈ A, a is a logical formula (atom).

I If ϕ is a logical formula, then so is ¬ϕ (negation).

I If ϕ and ψ are logical formulas, then so are (ϕ ∨ ψ) (disjunction) and (ϕ ∧ ψ) (conjunction).

Syntactical Conventions for Propositional Logic

Abbreviations:

I (ϕ → ψ) is short for (¬ϕ ∨ ψ) (implication)

I (ϕ ↔ ψ) is short for ((ϕ → ψ) ∧ (ψ → ϕ)) (equijunction)

I parentheses omitted when not necessary:

I

(¬) binds more tightly than binary connectives

I

(∧) binds more tightly than (∨), which binds more tightly than (→), which binds more tightly than (↔)

Semantics of Propositional Logic

Definition (Valuation, Model)

A valuation of propositions A is a function v : A → {T, F}.

Define the notation v | = ϕ (v satisfies ϕ; v is a model of ϕ;

ϕ is true under v ) for valuations v and formulas ϕ by

I v | = >

I v 6| = ⊥

I v | = a iff v (a) = T (for all a ∈ A)

I v | = ¬ϕ iff v 6| = ϕ

I v | = (ϕ ∨ ψ) iff (v | = ϕ or v | = ψ)

Propositional Logic Terminology (1)

I A logical formula ϕ is satisfiable

if there is at least one valuation v such that v | = ϕ.

I Otherwise it is unsatisfiable.

I A logical formula ϕ is valid or a tautology if v | = ϕ for all valuations v .

I A logical formula ψ is a logical consequence of a logical formula ϕ, written ϕ | = ψ, if v | = ψ for all valuations v with v | = ϕ.

I Two logical formulas ϕ and ψ are logically equivalent,

written ϕ ≡ ψ, if ϕ | = ψ and ψ | = ϕ.

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Propositional Logic Terminology (2)

I A logical formula that is a proposition a or a negated

proposition ¬a for some atomic proposition a ∈ A is a literal.

I A formula that is a disjunction of literals is a clause.

This includes unit clauses ` consisting of a single literal and the empty clause ⊥ consisting of zero literals.

I A formula that is a conjunction of literals is a monomial.

This includes unit monomials ` consisting of a single literal and the empty monomial > consisting of zero literals.

Normal forms:

I negation normal form (NNF)

I conjunctive normal form (CNF)

I disjunctive normal form (DNF)

A3. Transition Systems and Propositional Logic Summary

A3.4 Summary

A3. Transition Systems and Propositional Logic Summary

A3. Transition Systems and Propositional Logic

Planning and Optimization

A3. Transition Systems and Propositional Logic

Gabriele R¨ oger and Thomas Keller

Universit¨ at Basel

October 1, 2018

Planning and Optimization

October 1, 2018 — A3. Transition Systems and Propositional Logic

A3.1 Transition Systems A3.2 Example: Blocks World

A3.3 Reminder: Propositional Logic A3.4 Summary

Content of this Course

Planning

Classical

Tasks Progression/

Regression Complexity Heuristics

Probabilistic

MDPs Uninformed Search

Heuristic Search Monte-Carlo

Methods

Goals for Today

Today:

I introduce a mathematical model for planning tasks:

transition systems Chapter A3

I introduce compact representations for planning tasks suitable as input for planning algorithms

Chapter A4

A3.1 Transition Systems

Transition System Example

Transition systems are often depicted as directed arc-labeled graphs with decorations to indicate the initial state and goal states.

`

`

`

`

`

`

`

`

`

`

`

`

`

`

c(`

) = 1, c(`

) = 1, c (`

) = 5, c(`

) = 0

Transition Systems

Definition (Transition System)

A transition system is a 6-tuple T = hS , L, c, T , s 0 , S ? i where

I S is a finite set of states,

I L is a finite set of (transition) labels,

I c : L → R + 0 is a label cost function,

I T ⊆ S × L × S is the transition relation,

I s 0 ∈ S is the initial state, and

I S ? ⊆ S is the set of goal states.

We say that T has the transition hs, `, s 0 i if hs , `, s 0 i ∈ T .

Deterministic Transition Systems

Definition (Deterministic Transition System)

A transition system is called deterministic if for all states s

and all labels `, there is at most one state s 0 with s − → ` s 0 .

Example: previously shown transition system

Transition System Terminology (1)

We use common terminology from graph theory:

I s 0 successor of s if s → s 0

I s predecessor of s 0 if s → s 0

Transition System Terminology (2)

We use common terminology from graph theory:

I s 0 reachable from s if there exists a sequence of transitions s 0 − ` →

s 1 , . . . , s n−1 − ` →

s n s.t. s 0 = s and s n = s 0

Note: n = 0 possible; then s = s

s

, . . . , s

is called (state) path from s to s

`

, . . . , `

is called (label) path from s to s

s

−

A transition system is a 6-tuple T = hS , L, c, T , s ₀ , S _? i where

I c : L → R ⁺ ₀ is a label cost function,

I s ₀ ∈ S is the initial state, and

I S _? ⊆ S is the set of goal states.

We say that T has the transition hs, `, s ⁰ i if hs , `, s ⁰ i ∈ T .

and all labels `, there is at most one state s ⁰ with s − → ^` s ⁰ .

I s ⁰ successor of s if s → s ⁰

I s predecessor of s ⁰ if s → s ⁰

I s ⁰ reachable from s if there exists a sequence of transitions s ⁰ − ^` →

s ¹ , . . . , s ⁿ⁻¹ − ^` →

s ⁿ s.t. s ⁰ = s and s ⁿ = s ⁰

I s ⁰ reachable (without reference state) means reachable from initial state s ₀

I solution or goal path from s : path from s to some s ⁰ ∈ S _?

I transition system solvable if a goal path from s ₀ exists

I For this purpose, we will use propositional logic, which allows expressing information about 2 ⁿ states as logical formulas over n state variables.